We prove that quasi-trees of spaces satisfying the axiomatisation given by Bestvina, Bromberg and Fujiwara are quasi-isometric to tree-graded spaces in the sense of Dru\c{t}u and Sapir. As a corollary we deduce that mapping class groups quasi-isometrically embed into a finite product of tree-graded spaces. This gives bounds on compression exponent and guarantees finite Assouad-Nagata dimension. Moreover, using results of Buyalo and Masur-Minsky we prove that curve complexes embed into a finite product of trees, which we use to complete the theorem stated in the title."
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/3050